Projection and Contraction Methods for Solving Symmetric Cone Variational Inequalities

Abstract

In this paper, we employ projection and contraction methods (PC method) to solve variational inequality defined on a closed and convex symmetric cone (SCVI). By introducing Jordan product operator into a finite-dimensional inner product space $V$, we get an Euclidean Jordan algebras $(V, \circ)$. For a given $x\in V$, we have a spectral decomposition of $x$ with respect to a Jordan frame of $V$. Therefore, we can easily obtain projection of $x$ on the square cone $K$ of $V$($K$ is a symmetric cone). We describe some implementation issues of PC-methods for solving $\mbox{SCVI}(K, F)$ with uniformly strong monotone mapping $F$, while $K=R_+^n$, $K=\Lambda^n_+$ and $K=S^n_+$, respectively. Finally, we propose an implementation of PC methods for $\mbox{SCVI}(S^n_+, F)$ with uniformly strong monotone and Lipschitz continue mapping $F: S^n\right arrow S^n$, and give some numerical results to show validity of the proposed method.